微積分原理（上）

目錄
第1 章 實數集與初等函數··················1
1.1 實數集····································1
1.1.1 集合及其運算························1
1.1.2 映射···································3
1.1.3 可數集································3
1.1.4 實數集的性質························5
1.1.5 戴德金原理···························8
1.1.6 確界原理·····························8
習題1.1····································.10
1.2 初等函數······························.11
1.2.1 函數的概念························.11
1.2.2 函數的一些特性··················.12
1.2.3 函數的運算························.13
1.2.4 基本初等函數·····················.14
1.2.5 反函數及其存在條件·············.18
1.2.6 反三角函數························.19
*1.2.7 雙曲函數和反雙曲函數··········.22
*1.2.8 雙曲函數與三角函數之間
的聯系·····························.24
習題1.2····································.24
第2 章 數列極限····························.27
2.1 數列極限的概念·····················.27
習題2.1····································.30
2.2 數列極限的性質·····················.31
習題2.2····································.35
2.3 幾類特殊的數列·····················.36
2.3.1 無窮大數列與無窮小數列·······.36
2.3.2 無窮大數列與無界數列··········.36
2.3.3 Stolz 定理·························.38
習題2.3 ···································.40
2.4 實數連續(xù)性定理····················.41
2.4.1 單調有界定理·····················.41
2.4.2 閉區(qū)間套定理·····················.43
2.4.3 Bolzano-Weierstrass 定理·········.44
2.4.4 柯西收斂準則·····················.45
*2.4.5 有限覆蓋定理·····················.47
*2.4.6 聚點定理··························.48
習題2.4 ···································.48
*2.5 上極限與下極限···················.50
習題2.5 ···································.54
第3 章 函數極限與連續(xù)··················.55
3.1 函數極限的概念····················.55
3.1.1 函數在一點的極限···············.55
3.1.2 函數在無窮遠處的極限··········.58
習題3.1 ···································.58
3.2 函數極限的性質及運算···········.59
3.2.1 函數極限的性質··················.59
3.2.2 函數極限的四則運算·············.60
3.2.3 復合函數的極限··················.62
習題3.2 ···································.62
3.3 函數極限的存在條件··············.63
3.3.1 函數極限與數列極限的關系·····.63
3.3.2 兩個重要極限·····················.65
3.3.3 無窮大量與無窮小量·············.67
3.3.4 等價無窮小量代換求極限········.69
習題3.3····································.71
3.4 函數的連續(xù)···························.73
3.4.1 函數連續(xù)的概念··················.73
3.4.2 間斷點及其分類··················.74
3.4.3 連續(xù)函數的局部性質·············.78
習題3.4····································.78
3.5 閉區(qū)間上連續(xù)函數的性質········.79
3.5.1 閉區(qū)間上連續(xù)函數的基本性質··.79
3.5.2 反函數的連續(xù)性··················.82
3.5.3 一致連續(xù)性························.83
習題3.5····································.86
第4 章 導數與微分·························.89
4.1 導數的概念···························.89
4.1.1 導數概念的引出··················.89
4.1.2 函數可導的條件與性質··········.91
習題4.1····································.92
4.2 求導法則······························.94
4.2.1 導數的四則運算法則·············.94
4.2.2 反函數求導法則··················.96
4.2.3 復合函數的導數——鏈式法則···.97
4.2.4 隱函數求導法則··················.98
4.2.5 參數方程求導法則················.99
習題4.2····································102
4.3 函數的微分···························103
4.3.1 可微的概念························103
4.3.2 可微與可導的關系················104
4.3.3 微分在函數近似計算中的應用··105
4.3.4 微分的運算法則··················106
習題4.3····································106
4.4 高階導數與高階微分··············106
4.4.1 高階導數·························.107
4.4.2 高階微分·························.109
4.4.3 復合函數的微分·················.109
習題4.4 ··································.110
第5 章 微分學基本定理及應用········.112
5.1 微分中值定理······················.112
5.1.1 極值的概念與費馬定理·········.112
5.1.2 微分中值定理····················.113
習題5.1 ··································.118
5.2 洛必達法則··························.120
5.2.1 0
0
型不定式極限·················.120
5.2.2 ∞
∞
型不定式極限················.123
5.2.3 其他類型不定式極限············.125
習題5.2 ··································.126
5.3 泰勒公式及應用···················.127
5.3.1 泰勒公式·························.128
5.3.2 基本初等函數的展開式·········.130
5.3.3 泰勒公式的應用·················.134
習題5.3 ··································.138
5.4 單調性與極值······················.140
5.4.1 函數的單調性····················.140
5.4.2 函數取極值的條件··············.142
習題5.4 ··································.145
5.5 函數的凸性與函數作圖··········.147
5.5.1 函數的凸性······················.147
5.5.2 曲線的漸近性····················.152
5.5.3 函數作圖·························.153
習題5.5 ··································.155
*5.6 方程求根的牛頓迭代公式·····.155
第6 章 不定積分···························.160
6.1 原函數與不定積分················.160
6.1.1 原函數與6